COT3100.01, Fall 2002
S. Lang
(10/31/2002)
Answer key to Test #2
Nov. 4, 2002
Part I.
(39 pts., 3 pts. each) True/False, or shortanswer, questions. (No explanation needed and no
partial credit given.)
1.
Let
N
denote the set of natural numbers {0, 1, 2, …}, and let
A
⊆
N
and
B
⊆
N
.
Suppose set
A
contains a
smallest value and set
B
contains a smallest value.
Thus, the set
A
∩
B
contains a smallest value. (
False
,
since the set
A
∩
B
may be empty)
2.
Suppose
A
and
B
are both finite sets.
If
A
⊆
B
and 
A
 = 
B
, then
A
=
B
. (
True
, since
A
⊆
B
,
B
=
A
∪
(
B
–
A
) is true, which implies 
B
 = 
A
 + 
B
–
A
 because
A
∩
(
B
–
A
) =
∅
; thus, 
B
–
A
 = 0 because 
A
 = 
B

by assumption.
Therefore, the set
B
–
A
=
∅
, so
A
=
B
is true since
A
⊆
B
.)
3.
Suppose
n
≥
1 is an integer.
Then 2
n
>
C
(
n
,
k
) for any integer
k
with
n
≥
k
≥
1.
(
True
, by the formula
2
n
=
C
(
n
, 0) +
C
(
n
, 1) + …+
C
(
n
,
n
) >
C
(
n
,
k
) for any
k
,
n
≥
k
≥
1.)
4.
If
C
(
n
, 5) =
C
(
n
, 3), then the exact numerical value of
n
= 8, because
C
(8, 5) =
C
(8, 8 – 5) =
C
(8, 3).
5.
Suppose
n
≥
1 is an integer, and 1 + 2 + … +
n
= 20100.
The exact numerical value of
n
= 200, because
by the formula 1 + 2 + … +
n
=
2
)
1
(
+
n
n
= 20100,
n
(
n
+ 1) = 20100
⋅
2 = 200
⋅
201.
6.
In how many ways can 10 movie tickets (to the same movie) be divided between 3 persons where each
get at least one ticket?
C
(9, 2), by lining up the 10 tickets in a row which shows 9 gaps, and choosing 2
out of 9 divides the 10 tickets into 3 groups corresponding to ways of dividing the tickets to 3 persons.
(This is the same as the example of dividing 10 apples among 3 persons, as in the course notes.)
7.
In how many ways can the 9 letters in the word “PAPERTAPE” (and only these letters) be rearranged?
!
2
!
2
!
3
!
9
, because there are 9 letters for the permutations, consisting of 3 Ps, 2 As, 2 Es, and 1 of each
remaining letters R and T.
8.
A pocket contains 5 coins: 2 quarters, 2 dimes, and 1 penny.
When blindly picking two coins out of the
pocket, is it more likely to get 1 quarter and 1 dime, or more likely to get 1 penny and 1 dime?
Getting 1
quarter and 1 dime is more likely because there are 2
⋅
2 = 4 many ways to select 1 quarter and 1 dime,
compared to 1
⋅
2 = 2 ways to select 1 penny and 1 dime.
9.